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_Cases:

_{No repetition of 2:}

$cases = {5!}$

_{2 is present two times:}

$cases = {{5!} \over 2!}$

_{2 is present three time:}

$cases = {{5!} \over 3!}$

_{2 is present four time:}

$cases = {{5!} \over 4!}$

_{2 is present five time:}

$cases = {{5!} \over 5!}$

_{all cases:}

$total = {5!+{{5!} \over 2!}+{{5!} \over 3!}+{{5!} \over 4!}+{{5!} \over 5!}}$

$total = {206}$

A computer code calculates a total of 501 distinct permutations and begins with:

{{2, 2, 2, 2, 2}, {2, 2, 2, 2, 3}, {2, 2, 2, 2, 4}, {2, 2, 2, 2, 7}, {2, 2, 2, 2, 8}, {2, 2, 2, 3, 2}, {2, 2, 2, 3, 4}, {2, 2, 2, 3, 7}, {2, 2, 2, 3, 8}, {2, 2, 2, 4, 2}, {2, 2, 2, 4, 3}, {2, 2, 2, 4, 7}, {2, 2, 2, 4, 8}, {2, 2, 2, 7, 2}, ...........and ends with:

 {8, 4, 7, 3, 2}, {8, 7, 2, 2, 2}, {8, 7, 2, 2, 3}, {8, 7, 2, 2, 4}, {8, 7, 2, 3, 2}, {8, 7, 2, 3, 4}, {8, 7, 2, 4, 2}, {8, 7, 2, 4, 3}, {8, 7, 3, 2, 2}, {8, 7, 3, 2, 4}, {8, 7, 3, 4, 2}, {8, 7, 4, 2, 2}, {8, 7, 4, 2, 3}, {8, 7, 4, 3, 2} = 501 permutations.

OK, young person! If you don't believe it, here are the 16 distinct COMBINATIONS, which you must permute as follows:

[(2, 2, 2, 2, 2), (2, 2, 2, 2, 3), (2, 2, 2, 2, 4), (2, 2, 2, 2, 7), (2, 2, 2, 2, 8), (2, 2, 2, 3, 4), (2, 2, 2, 3, 7), (2, 2, 2, 3, 8), (2, 2, 2, 4, 7), (2, 2, 2, 4, 8), (2, 2, 2, 7, 8), (2, 2, 3, 4, 7), (2, 2, 3, 4, 8), (2, 2, 3, 7, 8), (2, 2, 4, 7, 8), (2, 3, 4, 7, 8)] >>Total = 16 distinct combinations.

[1 + 5*4 + 6*20 + 4*60 + 1*120] =501 distinct permutations.